Locus of mid points of the chords of the parabola $y^{2}=4ax$ which touch the circle $x^{2}+y^{2}=a^{2}$ is
- $(y^{2}-2a\mathrm{x})^{2}=a^{4}(y^{2}+4a^{2})$
- $(y^{2}-2a\mathrm{x})^{2}=a^{2}(y^{2}+4a^{2})$
- $(y^{2}-2a\mathrm{x})^{2}=2a^{4}(y^{2}+4a^{2})$
- $(y^{2}-2a\mathrm{x})^{2}=4a^{2}(y^{2}+4a^{2})$
The locus of the midpoint (h, k) of a chord of y^2=4ax is given by T=S1, which is ky - 2a(x+h) = k^2 - 4ah. This simplifies to ky = 2ax + k^2 - 2ah. The distance from the origin to this line must equal the radius 'a' of the circle x^2+y^2=a^2. Using the distance formula |k^2 - 2ah| / sqrt(k^2 + 4a^2) = a leads to the given equation.
Let the midpoint of the chord of the parabola y^2 = 4ax be (h, k). The equation of the chord with midpoint (h, k) is T = S1, which gives ky - 2a(x + h) = k^2 - 4ah, or 2ax - ky + k^2 - 2ah = 0. Since this chord touches the circle x^2 + y^2 = a^2, its distance from the origin (0, 0) must equal the radius a. Setting the distance equal to the radius gives |k^2 - 2ah| / sqrt((2a)^2 + (-k)^2) = a. Squaring both sides yields (k^2 - 2ah)^2 = a^2(4a^2 + k^2). Replacing h with x and k with y gives the locus (y^2 - 2ax)^2 = a^2(y^2 + 4a^2).